1. Field of the Invention
The present invention relates to a measurement method for performing precise measurement of the film thickness, the optical constants, and the like, for a thin film or an extremely-thin-film on a substrate, by analyzing the data acquired from a spectroscopic ellipsometer using the Best Local Minimum Calculation (which will be referred to as “BLMC” hereafter).
Furthermore, the present invention relates to an analysis method for an extremely-thin-film-double-layer structure by analyzing the data acquired from a spectroscopic ellipsometer using the Extended Best Local Minimum Calculation (which will be referred to as “EBLMC” hereafter).
Furthermore, the present invention relates to a analysis method for a thin-film-multi-layer structure using a spectroscopic ellipsometer. More specifically, the present invention relates to an analysis method for analyzing the data, which has been acquired from a spectroscopic ellipsometer, with regard to a multi-layer structure formed of unknown materials using the EBLMC as analysis means.
2. Description of the Related Art
(Description of General Background with Regard to a Spectroscopic Ellipsometer)
In general, a spectroscopic ellipsometer has a function wherein polarized light is cast onto a sample so as to measure the change in polarization between the incident light and the reflected light. Making an assumption that the complex refractive indexes of the ambient atmosphere and the substrate are known, the film thickness (d) and the complex refractive index (N=n−ik) can be calculated based upon the change in polarization between the incident light and the reflected light (in a case of analyzing a sample formed of only the substrate, the complex refractive index (N0) is calculated). The change in polarization (ρ) is represented by ρ=tan(ψ)exp(iΔ), and is dependent upon parameters such as the wavelength (λ), the incident angle (φ), the film thickness, the complex refractive indexes of the film, the substrate, the ambient atmosphere, and the like. The film thickness and the complex refractive index of the film of interest are obtained based upon the measured change in polarization and the following relationship.(d, n, k)=F(ρ)=F(ψ(λ,φ), Δ(λ,φ))
In case of single wavelength ellipsometer, if the incident angle is fixed, only two independent variables of three unknown values of (d, n, k) can be measured, and accordingly, there is the need to fix one of d, n, and k as a known value. Note that in the event that measurement is made with multiple incident angles, the number of measured variables increases, even if the single wavelength ellipsometer is used. However, measured pairs of (Ψ(φ1), Δ(φ1)) corresponding to different incidence angles (φ), are partly correlated, leading to difficulties in obtaining precise values of d,n, and k.
The measured spectrum measured by spectroscopic ellipsometer (ΨE(λi), ΔE(λi)), which represents the change in polarization due to reflection from single-layer or multi-layer thin films formed on a substrate, includes all information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer. However, the single combination of the information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer, cannot be simply extracted from the aforementioned measured spectra (excluding the case of semi-infinite substrate). In general, the method for extracting of the aforementioned single combination is referred to as “spectroscopic ellipsometry data analysis”. During this analysis, modeling is performed using the information with regard to n and k of the aforementioned substrate, and d, n, and k of each layer. The information regarding to n and k of the substrate and each layer included in the model is obtained from reference data (known table data), a dispersion formula, or optical constants of a single-layer thin film from a similar material.
The dispersion formula represents the wavelength-dependency of the dielectric constant of the material, wherein the dielectric constant ε(λ) can be determined in the optical range between near infrared light and ultraviolet light based upon the atomic structure of the material. Known examples of dispersion formulas include a formula based on classical physics (a harmonic oscillator), a formula based on quantum mechanics, an empirical formula, and the like, which generally include two or more parameters. The model is applied to the measured data by adjusting all the unknown values (thickness of each layer, parameters of the dispersion formula, volume fractions of material's components, or the like) included in the aforementioned model. This processing is referred to as “fitting”, wherein the thickness, parameters of dispersion formula, the volume fractions, and the like, of each layer are obtained. The complex dielectric constant ε(λ) of the material can be calculated from the parameters of the dispersion formula, based upon the fitting results. The relation between the complex dielectric constant of the material and the complex refractive index is represented by the following expression.ε(λ)=N2(λ)
Now, brief description will be made regarding fitting operation frequently employed in methods according to the present invention.
(Description Regarding the Fitting Figure of Merit χ2)
With the set of N pairs of measured (experimental) data as Exp(i=1, 2, and so on through N), and with the set of N pairs of the data calculated using the model as Mod(i=1, 2, and so on through N), making assumption that error of measurement follows normal distribution, and with the standard deviation as σi, the mean square error (χ2) is represented by the expression
      χ    2    =            [              1        /                  (                                    2              ⁢              N                        -            P                    )                    ]        ⁢                  ∑                  i          =          1                N            ⁢                                    (                                          Exp                i                            -                              Mod                i                                      )                    2                /                  σ          i          2                    wherein P represents the number of the parameters. The aforementioned expression indicates that the smaller χ2 is, the better the model matches the measured results. Accordingly, the best model can be selected from multiple models by selecting the model having the smallest χ2.
In a case of a sample wherein a single film is formed on a substrate, the change in polarization is proportional to the phase angle (β)×the cross-section area of the beam. The phase angle (β)(Film Phase Thickness) is represented by the following expression.β=2π(d/λ)(N2−NA2 sin2 φ)1/2 Making an assumption that the beam's cross section is constant, the change in polarization can be expressedChange in polarization ∝ Film thickness (d)×f(NA, N0, N, φ)
Here, NA denotes the complex refractive index of the ambient atmosphere, N0 denotes the complex refractive index of the substrate, N denotes the complex refractive index of the film, and φ denotes the incident angle. Note that in general, NA denotes the complex refractive index of the air, and accordingly, NA will be omitted hereafter. In the event that both the film thickness (d) and the complex refractive index (N) are small, the change in the phase angle (β) exhibits small value, in some cases, leading to difficulty in measurement. Specifically, in this case, the film thickness (d) and the complex refractive index (N) become strongly correlated.
Analysis of an extremely-thin-film-multi-layer structure is even more problematic, because the strong correlation between the film thickness (d) and the complex refractive index (N) may occur for each layer. In this case, it is difficult to obtain d, n, and k, for each film based upon the measurement results (ψE(λi), ΔE(λi)) which represent the change in polarization between the incident light and the reflected light.
Furthermore, as can be understood from the aforementioned expression, the precision of the incident angle affects the change in polarization. Accordingly, a method for obtaining a precise incident angle is necessary. That is to say, determination of the precise incident angle allows the precise determination of the change in the polarization of reflected light.
In the present invention, Effective Medium Theory (EMT) is used to calculate the effective dielectric function of materials, those dielectric function's wavelength dependence is difficult or impossible to express, using only one dispersion formula.
In general, the effective dielectric constant (ε) of the host material which contains N number of inclusions (guest materials), each inclusion is big enough to possess it's own dielectric constant, is represented by the expression
            (              ɛ        -                  ɛ          h                    )        /          (              ɛ        +                  k          ⁢                                          ⁢                      ɛ            h                              )        =            ∑              j        =        1            N        ⁢                            f          j                ⁡                  (                                    ɛ              j                        -                          ɛ              h                                )                    /              (                              ɛ            j                    +                      k            ⁢                                                  ⁢                          ɛ              h                                      )            wherein εh represents the dielectric constant of the host material, εj represents the dielectric constant of the j-th guest material, and k represents a screening factor. Now, let us consider a case in which one cannot distinguish between the host material and the guest material, i.e., a case that materials of comparable amount have been mixed. In this case, approximation can be made wherein the dielectric constant of the host material and the effective dielectric constant of mixed material are the same εh=ε, therefore εh in the aforementioned expression is replaced by the effective dielectric constant ε. The aforementioned approximation is called “Bruggeman Effective Medium Approximation”, which will be simply referred to as “EMA” in this specification hereafter. Using the EMA, the effective dielectric constant ε of a material, wherein three spherical components a, b, and c have been uniformly mixed, is obtained by the expressionfa(εa−ε)/(εa+2ε)+fb(εb−ε)/(εb+2ε)+fc(εc−ε)/(εc+2ε)=0wherein ε represents the effective dielectric constant which is to be obtained, εa, εb, and εc, represent the dielectric constants of the spherical components a, b, and c, respectively, and fa, fb, and fc, represent the volume fraction of the corresponding components. Volume fraction will be referred to as “Vf”, hereafter. Note that fa+fb+fc=1.
Effective Medium Approximation (EMA) is applicable, if the separate regions (components) of mixed material are small compared to the wavelength of light. EMA is used to model thin film on substrate, if this film is either microscopically inhomogeneous or discontinuous or formed by several physically mixed materials.
Now, description will be made regarding a case that the materials a, b, and c have been mixed. In this case, EMA is used to calculate the dielectric constant of the mixed layer from the volume fractions of each component and the dielectric constants of corresponding materials a, b and c. Dielectric constant of each component can be determined by either reference data or dispersion formula. Assuming the mixed layer thickness, model can be built and fitted to the measured data.
The calculation methods which are referred to as “BLMC (Best Local Minimum Calculation)” and “EBLMC (Extended Best Local Minimum Calculation) are frequently used in the analysis method according to the present invention. In this specification, these calculation methods will be referred to the abbreviations of “BLMC” and “EBLMC” hereafter.
BLMC is used for analyzing a single-layer structure. In the analysis using the BLMC, fitting is made with a predetermined procedure while adjusting the initial values of predetermined parameters within a certain range, so as to obtain the best results.
EBLMC is used for analyzing a multi-layer structure. In the analysis using the EBLMC, BLMC is repeated for a film of interest while maintaining the predetermined kinds of parameters of the other films to multiple values at and around the medians thereof, and the best results are determined as the results obtained with the EBLMC.